Summation of a slowly convergent fourier series occurring in a fluid motion problem

Abstract

Methods are described for the numerical evaluation of a slowly convergent sine series S ( 0 ) (0⩽0⩽1/2n), in which the coefficients satisfy a linear recurrence relation; a connexion is established between S ( 0 ) and the hypergeometric function. A general method is presented for the evaluation of an extensive class of power series in a complex variable, based on the asymptotic expansion of the coefficients in descending inverse factorials. The proof is given of a lemma on the asymptotic expansion of products of gamma functions, which is fundamental in the application of the method to hypergeometric series. The method is applied to S ( 0 ) in two ways: in the first, use is made of the connexion with the hypergeometric function; in the second, the general coefficient in the series is expanded directly by solving the recurrence relation in series. Another method of calculating S ( 0 ), founded on Euler’s transformation of series, can be used over a range given approximately by 0°⩽0⩽70°.